How to express 'y*log(1+(dx/(ax+by+c)))' and 'log(xy/(ax+by+c))' using CVX

[lxn]

Question: I want to know that how can we express the following expressions using CVX?

y*log(1+(dx/(ax+by+c))),
‘log(xy/(ax+by+c))’
where it is confirmed that this is a concave function for x>0, y>0. a, b, c and d are constant values.

It will be really appreciated if someone can address the above notes problem. Thanks!

[jackfsuia (Jack)]

Hi, lxn. The 1st expression can be reduced to this form

X*log((ay+x)/(cy+x)) a>c

[image] how to be expressed by cvx？

the 2nd expression can be reduced to -log(a/y+b/x+c/(xy)), equivalently, -log(a/y+b/x+c/(e^z)) and z<=logx+logy, which can be further addressed by

How to express log(1+ sum ( 1./x ) ) in CVX?

Use the first approach in section 5.2.7 Log-sum-inv of https://docs.mosek.com/modeling-cookbook/expo.html#exponential-cone 4 by using CVX’s log_sum_exp for the ti >= … constraint, and also include the xi >= … constraint in that link as written. Prepend your x with a new first element of 1 to handle the `1+``

[lxn]

I’m very sorry that I still don’t know how to express the two expressions. Can you be more detailed? Many thanks for your kind response.